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Question
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
Options
0°
45°
90°
none of these
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Solution
90°
\[\text { Given }: \]
\[xy = a^2 . . . \left( 1 \right)\]
\[ x^2 - y^2 = 2 a^2 . . . \left( 2 \right)\]
\[\text { Let} \left( x_1 , y_1 \right)\text {be the point of intersection }.\]
\[\text { On differentiating (1) w.r.t. x, we get }\]
\[x\frac{dy}{dx} + y = 0\]
\[ \Rightarrow \frac{dy}{dx} = \frac{- y}{x}\]
\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = \frac{- y_1}{x_1}\]
\[\text { On differentiating (2) w.r.t.x, we get }\]
\[2x - 2y \frac{dy}{dx} = 0\]
\[ \Rightarrow \frac{dy}{dx} = \frac{x}{y}\]
\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = \frac{x_1}{y_1}\]
\[\text { Now,} \]
\[ m_1 \times m_2 = \frac{- y_1}{x_1} \times \frac{x_1}{y_1}\]
\[ \Rightarrow m_1 \times m_2 = - 1\]
\[ \because m_1 \times m = - 1\]
\[\text { So, the angle between the curves is } 90°\]
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